1. (15 points)

The linear combination of vectors

$v^T=[1,2,0]$ and

$w^T=[0,0,1]$ fill a

$plane$ in

$R^3$.

1. Describe that plane with in matrix form.
2. Find one vector NOT that plane.
3. Find a vector perpendicular to that plane.

2. (10 points)

I. A vector

$v=(1,1,\dots ,1)$ in 9-dimensions, please find a unit vector

$\hat{v}$ in the same direction as

$v$.
II. Find another unit vector

$\hat{w}$ that is perpendicular to

$v$.

3. (40 points) A system of linear equation as:

$$\begin{array}{rl}{x}_1+x_2+x_3& =7\\ x_1+x_2-x_3& =5\\ x_1-x_2+x_3& =3\end{array}$$

1. Use elimination to make
   $PA\to U$ and back substitution to solve
   $x$.
2. Factorize
   $PA=LU$.
3. Compare the multiplier of each elimination step and lower triangle matrix
   $L$.
4. Using Gauss-Jordan to find
   $A^{-1}$ and make
   $x=A^{-1}b$.

4. (20 points)

If a

$3\times 3$ matrix

$A$ conducts this operation: row1 + row2 = row3,

• Is
  $A$ singular or invertible?
• Find a nonzero solution of
  $Ax=[0,0,0{]}^T$.

5. (15 points)

• The row vector
  $x^T$ times
  $A$ times the column
  $y$ produces what number?

$$x^{T}Ay=\left[\begin{array}{cc}0& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right]\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]$$

• This is the row
  $x^{T}A=\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}$ times the column
  $y$.
• This is the row
  $x^T=[0,1]$ times the column
  $Ay=\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}$.